Multivariate data vs. multivariate analysis
We’ve already seen multivariate data in multiple regression and multi-factor ANOVA, but now we’ll look at cases with multiple response variables.
Functional methods: - Clear response and predictor variables - Goal: relate Y’s to X’s - Examples: MANOVA, PERMANOVA
Structural methods: - Find patterns/structure in data - Often no clear predictors - Examples: PCA, NMDS, Cluster Analysis
Scaling/Ordination Methods: - Reduce dimensions with new derived variables - Summarize patterns in data - Examples: PCA, CCA
Dissimilarity-Based Methods: - Measure dissimilarity between objects - Visualize relationships between objects - Examples: NMDS, Cluster Analysis
Key concept
Eigenvalues (λ) represent the amount of variation explained by each new derived variable, while eigenvectors contain the coefficients showing how original variables contribute to each component.
Transformations: - Log transformation for skewed data - Root transformations for count data - Fourth-root for species abundance data
Standardization: - Centering: subtract mean (mean = 0) - Standardization: divide by SD (SD = 1) - Crucial for variables with different units - May not be appropriate for species data
Why standardize?
Standardization ensures all variables contribute equally to the analysis regardless of their original units or scales of measurement. Without it, variables with larger values or variances would dominate the results.
Multivariate Outliers: - Objects with unusual patterns across variables - Detected with Mahalanobis distance (d²) - Test against χ² distribution with p df
Missing Observations: - Common approaches: - Deletion: remove affected object or variable - Imputation: estimate missing values - Maximum likelihood methods - Multiple imputation
MANOVA Assumptions
Multivariate data requires special techniques to account for correlations between variables
Functional methods (MANOVA) test hypotheses about group differences
Structural methods (PCA, NMDS) find patterns in data
Distance measures quantify similarities between objects
Data standardization is crucial for variables with different units
Multivariate graphics help visualize complex relationships